Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

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Numerical solution of saddle point problems

https://www.southampton.ac.uk/~nwb/lectures/GoodPracticeCFD/Articles/Validation_SAND2002-0529.pdf

Verification and Validation in Computational Fluid Dynamics

This article reviews linear instability analysis of flows over or through complex two-dimensional (2D) and 3D geometries. In the three decades since it first appeared in the literature, global instability analysis, based on the solution of the multidimensional eigenvalue and/or initial value problem, is continuously broadening both in scope and in depth. To date it has dealt successfully with a wide range of applications arising in aerospace engineering, physiological flows, food processing, and nuclear-reactor safety. In recent years, nonmodal analysis has complemented the more traditional modal approach and increased knowledge of flow instability physics. Recent highlights delivered by the application of either modal or nonmodal global analysis are briefly discussed. A conscious effort is made to demystify both the tools currently utilized and the jargon employed to describe them, demonstrating the simplicity of the analysis. Hopefully this will provide new impulses for the creation of next-generation algorithms capable of coping with the main open research areas in which step-change progress can be expected by the application of the theory: instability analysis of fully inhomogeneous, 3D flows and control thereof.

Global Linear Instability

A high resolution scheme with improved iterative convergence properties was devised by incorporating total-variation diminishing constraints, appropriate for unsteady problems, into an implicit time-marching method used for steady flow problems. The new scheme, referred to as Convergent and Universally Bounded Interpolation Scheme for the Treatment of Advection (CUBISTA), has similar accuracy to the well-known SMART scheme, both being formally third-order accurate on uniform meshes for smooth flows. Three demonstration problems are considered: (1) advection of three scalar profiles, a step, a sine-squared, and a semi-ellipse; (2) Newtonian flow over a backward-facing step; and (3) viscoelastic flow through a planar contraction and around a cylinder. For the case of the viscoelastic flows, in which the high resolution schemes are also used to represent the advective terms in the constitutive equation, it is shown that only the new scheme is able to provide a converged solution to the prescribed tolerance. Copyright © 2003 John Wiley & Sons, Ltd.

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A convergent and universally bounded interpolation scheme for the treatment of advection

Results are reported from a three-dimensional computational stability analysis of flow over a backward-facing step with an expansion ratio (outlet to inlet height) of 2 at Reynolds numbers between 450 and 1050. The analysis shows that the first absolute linear instability of the steady two-dimensional flow is a steady three-dimensional bifurcation at a critical Reynolds number of 748. The critical eigenmode is localized to the primary separation bubble and has a flat roll structure with a spanwise wavelength of 6.9 step heights. The system is further shown to be absolutely stable to two-dimensional perturbations up to a Reynolds number of 1500. Stability spectra and visualizations of the global modes of the system are presented for representative Reynolds numbers.

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Three-dimensional instability in flow over a backward-facing step

A detailed case study is made of one particular solution of the 2D incompressible Navier-Stokes equations. Careful mesh refinement studies were made using four different methods (and computer codes): (1) a high-order finite-element method solving the unsteady equations by time-marching; (2) a high-order finite-element method solving both the steady equations and the associated linear- stability problem; (3) a second-order finite difference method solving the unsteady equations in streamfunction form by time-marching; and (4) a spectral-element method solving the unsteady equations by time-marching. The unanimous conclusion is that the correct solution for flow over the backward-facing step at Re=800 is steady - and it is stable, to both small and large perturbations.

Is the steady viscous incompressible two‐dimensional flow over a backward‐facing step at Re = 800 stable?

Block matrices with a special structure arise from mixed finite element discretizations of incompressible flow problems. This paper is concerned with an analysis of the eigenvalue problem for such matrices and the derivation of two shifted eigenvalue problems that are more suited to numerical solution by iterative algorithms like simultaneous iteration and Arnoldi's method. The application of the shifted eigenvalue problems to the determination of the eigenvalue of smallest real part is discussed and a numerical example arising from a stability analysis of double-diffusive convection is described.

Eigenvalues of Block Matrices Arising from Problems in Fluid Mechanics

This paper is concerned with a numerical approach to the problem of finding the leftmost eigenvalues of large sparse nonsymmetric generalised eigenvalue problems which arise in stability studies of incompressible fluid flow problems. The matrices have a special block structure that is typical of mixed finite element discretizations for such problems. The numerical approach is an extension of the hybrid technique introduced by Saad [22] and utilizes the idea of preconditioning the eigenvalue problem before applying Arnoldi's method. Two preconditioners, one a modified Cayley transform, the other a Chebyshev polynomial transform, are compared in numerical experiments on a double diffusive convection problem and the Cayley transform proves superior. The Cayley transform is then used to provide numerical results for the finite Taylor problem.

Eigenvalues of the discretized Navier-Stokes equation with application to the detection of Hopf bifurcations

This paper is concerned with the detection of Hopf bifurcations in the parameter dependent nonlinear system 

$$\frac{{dx}}{{dt}} = f\left( {x,\lambda } \right),f:{R^n} \times R \mapsto {R^n},x \in {R^n},\lambda \in R.$$

(1)

The set $\Gamma : = \left\{ {\left( {x,\lambda } \right) \in {R^{n + 1}}:f\left( {x,\lambda } \right) = 0} \right\}$ represents the steady state solutions of (1) and it is often important to determine the (linearised) stability of a branch of Γ. If µ i denotes the eigenvalue with largest real part of the Jacobian matrix A:= fx(x,λ), then a steady state solution is stable (unstable) if Re(µ 1) is negative (positive). Also in applications it is desirable to detect a point of Γ where µ i is complex and Re(µ 1) changes sign as A varies, called a Hopf bifurcation point. If n is small in (1) it is certainly simplest and probably best to find all the eigenvalues of A using the QR algorithm. However for systems arising from spatial discretisations of p.d.e.’s n is typically very large with the Jacobian matrix sparse, and the direct application of the QR algorithm for a nonsymmetric matrix will be very expensive or may not even be feasible. Of course we need not find all the eigenvalues of A since only the eigenvalues which determine the stability are wanted. This paper is about the application of iterative methods for the calculation of the eigenvalues of A with largest real part.

Two Methods for the Numerical Detection of Hopf Bifurcations

During the computation of a one-parameter set of steady solutions to a time dependent problem, it is often of great interest to know when periodic behaviour can occur. This Hopf bifurcation is characterised by the linearisation about a steady state solution possessing a purely imaginary pair of eigenvalues. The cheap and reliable detection of such bifurcation is not however straightforward. The present paper examines the problem and suggests various strategies for solving it.

T. J. Garratt

Papers

Is the steady viscous incompressible two‐dimensional flow over a backward‐facing step at Re = 800 stable?

Eigenvalues of Block Matrices Arising from Problems in Fluid Mechanics

Eigenvalues of the discretized Navier-Stokes equation with application to the detection of Hopf bifurcations

Two Methods for the Numerical Detection of Hopf Bifurcations

The numerical detection of hopf bifurcation points